Strongly meager sets of size continuum

We will construct several models where there are no strongly meager sets of size 2⁰. First author partially supported by NSF grant DMS 0200671.Second author partially supported by Israel Science Foundation and NSF grant DMS 0072560. Publication 807. Mathematics Subject Classification (2000):03E15, 03E20     

Strongly meager and strong measure zero sets

 In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5>&ensp;03E35 RID="ID=" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658 Received: 6 January 1999 / Revised version: 20 July 1999 / Published online: 25 February 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5>&ensp;03E35 RID=" ID=" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658     

Strongly meager sets and their uniformly continuous images

We prove the following theorems:

(1) Suppose that is a continuous function and is a Sierpinski set. Then

(A)

for any strongly measure zero set , the image is an -set,

(B)

is a perfectly meager set in the transitive sense.

(2) Every strongly meager set is completely Ramsey null.

     

Strongly meager sets of real numbers and tree forcing notions

We show that every strongly meager set has the - and the - property.

     

Universally meager sets

We study category counterparts of the notion of a universal measure zero set of reals.

We say that a set is universally meager if every Borel isomorphic image of is meager in . We give various equivalent definitions emphasizing analogies with the universally null sets of reals.

In particular, two problems emerging from an earlier work of Grzegorek are solved.

     

On perfectly meager sets

We show that it is consistent that the product of perfectly meager sets is perfectly meager.

     

Perfectly meager sets and universally null sets

We will show that there is no example of a set distinguishing between universally null and perfectly meager sets.

     

Universally meager sets, II

We present new characterizations of universally meager sets, shown in [P. Zakrzewski, Universally meager sets, Proc. Amer. Math. Soc. 129 (6) (2001) 1793-1798] to be a category analog of universally null sets. In particular, we address the question of how this class is related to another class of universally meager sets, recently introduced by Todorcevic [S. Todorcevic, Universally meager sets and principles of generic continuity and selection in Banach spaces, Adv. Math. 208 (2007) 274-298].     

Saturation, Suslin trees and meager sets

We show, using a variation of Woodin’s partial order ℙ_max, that it is possible to destroy the saturation of the nonstationary ideal on ω₁ by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ_max variations where one does not try to arrange it otherwise. In the last section, we show that it is possible to have a nonmeager set of reals of size ℵ₁, saturation of the nonstationary ideal, and no weakly Lusin sequences, answering a question of Shelah and Zapletal.Supported by the Japan Society for the Promotion of Science, the Mittag-Leffler Institute and the São Paulo State Research Support Foundation (FAPESP, Grant # 02/11551-3).     

Strongly flow-invariant sets

Some previously established existence theorems for periodic solutions of nonlinear, two-diminsional systems contain a hypothesis which requires inversion of the given system and is generally difficult to establish. This difficulty is eliminated for certain subclasses of systems. Further, some results due to Cronin concerning the number of such solutions and the characterization of hyperbolic sectors about critical points are shown to apply to the systems considered     

On perfectly meager sets in the transitive sense

We prove that assuming one can always find a perfectly meager set, which is not perfectly meager in the transitive sense.

     

Strongly dominating sets of reals

We analyze the structure of strongly dominating sets of reals introduced in Goldstern et al. (Proc Am Math Soc 123(5):1573–1581, 1995). We prove that for every ${\kappa < \mathfrak{b}}$ κ < b a ${\kappa}$ κ -Suslin set ${A\subseteq{}^\omega\omega}$ A ? ω ω is strongly dominating if and only if A has a Laver perfect subset. We also investigate the structure of the class l of Baire sets for the Laver category base and compare the σ-ideal of sets which are not strongly dominating with the Laver ideal l ⁰.     

On graphs that can be oriented as diagrams of ordered sets

We study some equivalent and necessary conditions for a finite graph to be the covering graph of a (partially) ordered set. For each 1, M. Aigner and G. Prins have introduced a notion of a vertex colouring, here called -good colouring, such that a 1-good colouring is the usual concept and graphs that have a 2-good colouring are precisely covering graphs. We present some inequalities for the corresponding chromatic numbers , especially for x ₂. There exist graphs that satisfy these inequalities for =2 but are not covering graphs. We show also that x ₂ cannot be bounded by a function of x=x ₁. A construction of Neetil and Rödl is used to show that x ₂ is not bounded by a function of the girth.     

Strongly antiproximinal sets in Banach spaces

Translated from Matematicheskie Zametki, Vol. 47, No. 2, pp. 130–136, February, 1990.     

Strongly non-repetitive sequences and progression-free sets

By a simple method we show the existence of (1) a sequence on two symbols in which no four blocks occur consecutively that are permutations of each other, and (2) a sequence on three symbols in which no three blocks occur consecutively that are permutations of each other. The problem of the existence of a sequence on four symbols in which no two blocks occur consecutively that are permutations of each other remains open.     

Unique satisfiability of Horn sets can be solved in nearly linear time

If a Horn set I has a single satisfying truth assignment or model then that model is said to be unique for I. The question of determining whether a unique model exists for a given Horn set I is shown to be solved in O((L)*L) time, where L is the sum of the lengths of the clauses in I and is the inverse Ackermann function. It is also shown that if LA*log (A) where A is the number of distinct proposition letters then unique satisfiability can be determined in O(L) time.     

强拟闭集与强拟连续

利用半闭集引入强拟闭集概念,研究了半开集、强拟开集、强拟闭集概念之间的关系,得到了强拟闭集是连续闭映射下的不变量及其相关性质;最后给出强拟连续概念并得到其等价刻画.